One to two months before kidney transplant surgery, blood stem cells will be removed from the donor and the cells will be frozen. After transplant surgery, the recipient will receive radiation and anti-T cell antibody treatments for two weeks to prepare for injection of the stem cells. The stem cells will be injected at the end of the two-week treatment. If the stem cells persist in the recipient, immunosuppressive drugs will be gradually reduced until they are withdrawn completely at least six months after transplantation. Patients will be followed in the Stanford clinics for transplant patients. Patients who live outside of the San Francisco Bay Area must remain near Stanford for six weeks after transplant surgery.